Games with Exhaustible Resources - Semantic Scholar
Games with Exhaustible Resources Chris Harris∗
Sam Howison†
Ronnie Sircar‡
August 14, 2009
Abstract We study N -player continuous-time Cournot games in an oligopoly where firms choose production quantities. These are nonzero-sum differential games, whose value functions may be characterized by systems of nonlinear Hamilton-Jacobi partial differential equations. When resources are in finite supply, such as oil, exhaustibility enters as boundary conditions for the PDEs. We analyze the problem when there is an alternative, but expensive, technology (for example solar power for energy production), and give an asymptotic approximation in the limit of small exhaustibility. We illustrate the two-player problem by numerical solutions, and discuss the impact of limited oil reserves on production and oil prices in the duopoly case.
Contents 1 Introduction
2
2 Static Cournot Game 2.1 General Price Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Examples: Constant Prudence Price Curves . . . . . . . . . . . . . . . . . .
3 4 8
3 Differential Game & Exhaustibility 3.1 Dynamic Cournot Competition . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Exhaustibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Neumann Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . .
12 12 14 15
4 Small Exhaustibility Approximation 4.1 Static Game Small Cost Perturbation . . . . 4.2 Differential Game Small Cost Perturbation . 4.3 Time to Exhaustion in the Symmetric Game 4.4 Illustration: Duopoly . . . . . . . . . . . . .
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Faculty of Economics, Cambridge University, Sidgwick Avenue, Cambridge CB3 9DD; [email protected] † OCIAM, Mathematical Institute, Oxford University, 24-29 St. Giles’, Oxford OX1 3LB; [email protected] ‡ ORFE Department, Princeton University, Sherrerd Hall, Princeton NJ 08544; [email protected]. Work partially supported by NSF grant DMS-0807440.
1
5 Duopoly Extraction Problem with 5.1 The Axis Game & Blockading . . 5.2 Type of the PDEs . . . . . . . . . 5.3 Numerical Solutions . . . . . . . .
the Linear . . . . . . . . . . . . . . . . . . . . .
6 Conclusions
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Pricing . . . . . . . . . . . . . . .
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24 25 30 33 37
Introduction
The problem of dwindling oil reserves and its impact on energy supply and prices is of longstanding importance. One way to analyze the issues is to view energy markets as being governed by a small number of competitive firms or countries, that is as oligopolies, and to model the formation of prices and supplies within this competitive framework. Game theory provides a natural way to frame the outcome of competition within different choices of market mechanism. Exhaustibility, meanwhile, requires analysis of how anticipation of changing resources impacts production and prices, and therefore leads to dynamic games. Here we study continuous-time (or differential) games arising from competition over a resource in limited supply. The games are nonzero-sum as all players act to maximize their own profits. Typical models of industrial organization in the economics literature are restricted to the study of one or two-period games. For ordinary and stochastic zero-sum differential games, there is a fairly general theory [7, 11], including a viscosity theory for their associated (scalar) Hamilton-Jacobi-Bellman-Isaacs PDEs. In the nonzero-sum case where systems of equations for the value functions of all the players arise, there is, to our knowledge, no similar general theory. Some recent books that discuss nonzero-sum deterministic and stochastic differential games are [1, 6]. There is also a literature on exhaustibility (or capacity constraints), but not, primarily, in the context of continuous-time models. We mention [5] as a reference for the literature on exhaustible resources up through the 1970s. When the quantity being produced is in finite supply, such as oil, exhaustibility is a “game-changer”, and enters as boundary conditions for the PDEs. We analyze the problem when there is an alternative resource (for example solar technology for energy production), which is inexhaustible, but much more costly to produce than extracting oil. Of interest is the impact on oil extraction rates, and hence market prices, as reserves run out and energy production must switch to more expensive renewable sources. We begin by analysing the static N-player Cournot game as a function of the costs of production of the firms. When we move to the dynamic problem in Section 3, exhaustibility acts like a varying cost, that depends on the dynamic game’s value functions, in a static game at the infinitesimal level. Therefore, we devote some effort in Section 2 to establish existence and uniqueness of a Nash equilibrium for the static game with players who have different costs. The static game equilibrium production and profit functions are essential ingredients for the partial differential equations characterizing the dynamic game in Section 3. In Section 4, we derive a perturbation approximation for the case when the cost of the alternative technology is small. Section 5 studies in detail and with numerical PDE solutions a specific two-player example. In particular, we discuss the issue of one firm being blockaded out of competition by the dominance of the other. Section 6 concludes.
2
2
Static Cournot Game
In the bulk of the paper we will analyse a dynamic version of the Cournot model of competition: producers of the resource set quantities which they bring to the market, and the price is then determined from the total quantity produced. Before moving to the dynamic game with exhaustibility, we analyze the static (one-period, or stage) game and introduce notation from it that will be needed later. We take as given a price (or inverse-demand) function P : (0, ∞) → R that gives market price (per unit) as a function of quantity produced and put on the market. There are N ≥ 1 players. Each player i chooses a quantity qi ∈ [0, ∞) to produce at unit cost of production ai ≥ 0, and the market price is determined from the total production. In the original example described by Cournot [4], the inexhaustible resource being produced was mineral water. Once each player chooses his quantity, the market price is given by P (Q),
where Q =
N X
qj .
j=1
The profit of player i is the quantity he produces multiplied by price minus cost: qi (P (Q−i + qi ) − ai ) if qi > 0, (1) π(qi , Q−i , ai ) = 0 if qi = 0, P where Q−i = j6=i qj is total production by the players other than i. We allow for the possibility that P (0+ ) = +∞, but specify π(qi , Q−i , ai ) = 0 when qi = 0: if a player does not produce anything, then he makes no profit. Each player seeks to maximize his own profit, taking the quantities produced by the other players as given. More precisely: ∗ Definition 2.1. A Nash equilibrium is a vector q ∗ = (q1∗ , q2∗ , ..., qN ) ∈ [0, ∞)N such that, for all i, π(qi∗ , Q∗−i , ai ) = max π(qi , Q∗−i , ai ), (2) qi ∈[0,∞) P where Q∗−i = j6=i qj∗ . That is, each player’s equilibrium production qi∗ maximizes his own profit π(·, Q∗−i, ai ) when the other N − 1 players produce their equilibrium quantities. If, in addition, qi∗ > 0 for all i, then we call q ∗ an interior Nash equilibrium.
The main aim of this section is to show that, under suitable conditions on the price function P and the cost vector a = (a1 , a2 , · · · , aN ), a Nash equilibrium exists and is unique. Much of the literature on static Cournot games assumes identical costs among the players (so that all or none will participate), or assumes all costs are small enough that there is an interior Nash equilibrium. In the models we study here, costs are linear in quantity (constant marginal cost), but we shall need to consider the situation of hetergeneous costs among the players, which will be related to different proximities to exhaustion in the dynamic game of Section 3. An important issue arising from this is that it may be too costly for some players to participate. To our knowledge, this aspect has not been fully addressed before, even in the static game. We refer to [16, Chapter 4] for a discussion and references on general existence results for static Cournot games. Our primary assumptions on P are: 3
Assumption 2.2. The price function P is twice continuously differentiable, with P ′ < 0 everywhere on (0, ∞); and there exists η ∈ (0, ∞) such that P (η) = 0. The first part of this assumption is natural: the greater total production, the less the market will be willing to pay per unit. The second part implies that P (Q) < 0 for Q > η: if there is over-production, players have to pay to have the surplus removed. Nonetheless, it should be noted that negative prices will play no role in our analysis, because profitmaximizing players with positive costs will never produce at a level at which prices are negative. We shall refer to η as the saturation point. We order the firms by their costs and assume they are strictly less than the choke price P (0+ ): 0 ≤ a1 ≤ a2 ≤ ... ≤ aN < P (0+ ). (3)
When some firms have equal costs, the ordering is arbitrary and does not affect the analysis that follows. The assumption that ai < P (0+ ) for all i ensures that, in any Nash equilibrium P ∗ q ∗ , total production Q∗ = N i=1 qi will be strictly positive, for if all players other than i produce nothing, so that qj = 0 for all j 6= i, then player i can make a strictly positive profit qi (P (qi ) − ai ) by producing a small positive amount qi (and may be able to do much better). Having all players produce nothing is therefore not a Nash equilibrium. The assumption that ai ≥ 0 for all i ensures that P (Q∗ ) > 0. In particular, Q∗ < η. The behaviour of P is best characterized in terms of the relative prudence of P , namely ρ(Q) = −
Q P ′′ (Q) . P ′ (Q)
(4)
Our terminology here is adapted from [12]. We also define ρ = sup ρ(Q).
(5)
Q∈(0,∞)
We turn now to the detailed analysis of Nash equilibrium, first for general price functions P , and then, in Section 2.2, for a convenient family of examples which are amenable to explicit calculations.
2.1
General Price Functions
Suppose that q ∗ is an interior Nash equilibrium. Then, for all i ∈ {1, 2, ..., N}, qi∗ must satisfy the first-order condition 0=
∂π ∗ ∗ (q , Q , ai ) = qi∗ P ′ (Q∗−i + qi∗ ) + P (Q∗−i + qi∗ ) − ai . ∂qi i −i
(6)
Summing over i, we obtain 0 = Q∗ P ′ (Q∗ ) + N P (Q∗ ) − AN ,
(7)
P PN ∗ where Q∗ = Q∗−i + qi∗ = N i=1 qi is total production and AN = i=1 ai is the sum of the unit ∗ costs. In other words, Q must satisfy the scalar equation fN (Q) = AN , where fN (Q) = QP ′ (Q) + N P (Q), 4
Q > 0.
On the other hand, given a solution of this equation, making qi∗ the subject of equation (6), a candidate Nash equilibrium is P (Q∗ ) − ai . (8) qi∗ = −P ′ (Q∗ ) Using (6), we can also express the candidate equilibrium quantities in (8) as ! ∗ P (Q ) − a i qi∗ = PN Q∗ . ∗ j=1 (P (Q ) − aj )
This has the interpretation that, once the equilibrium total quantity Q∗ is determined, the fraction produced by player i is the deviation of his cost ai from the market price P (Q∗ ) relative to the total deviation of all players’ costs from the price. However, some qi∗ defined by (8) may be negative, and so we must also consider Nash equilibria in which some players do not produce. For 1 ≤ n ≤ N, we define fn (Q) = QP ′ (Q) + n P (Q), and An =
Pn
i=1
Q > 0,
(9)
ai . We then have the following lemma.
Lemma 2.3. Fix n ∈ {1, · · · , N}, and suppose that ρ < n + 1. Then there is a unique Q∗n ∈ (0, η) such that fn (Q∗n ) = An . Proof. It is sufficient to show that fn is decreasing with fn (0+ ) > An and fn (η) < 0, so there is a unique root of fn (Q) = An in (0, η). We compute fn′ (Q) = QP ′′ (Q) + (n + 1) P ′(Q) = (n + 1 − ρ(Q)) P ′ (Q) ≤ (n + 1 − ρ) P ′(Q).
(10)
Hence fn′ < 0 on (0, ∞). We also have fn (η) = ηP ′(η) < 0 (since P (η) = 0). We therefore need only show that fn (0+ ) > An . In the case where P (0+ ) is finite and P ′ (0+ ) exists and is finite, fn (0+ ) = nP (0+ ) > An , from (3). The case where the choke price is infinite (P (0+ ) = +∞) is a little more involved, and is handled in the appendix. Then, for each n > max(0, ρ − 1), we have the following n-player candidate Nash equilibrium: ( P (Q∗n )−ai for 1 ≤ i ≤ n, ∗ −P ′ (Q∗n ) (11) qi = 0 for n + 1 ≤ i ≤ N, where Q∗n is the unique solution of fn (Q) = An , and we recall that the players are ordered by their production costs ai . This candidate equilibrium can fail to be a Nash equilibrium of the game as a whole in one of three ways: 1. it may happen that qi∗ < 0 for some 1 ≤ i ≤ n; 2. it may happen that ai < P (Q∗n ) for some n + 1 ≤ i ≤ N; or 3. it may happen that, for some 1 ≤ i ≤ n, qi∗ is not a global maximum of π(·, Q−i , ai ). 5
The first case occurs if and only if ai > P (Q∗n ), that is, the unit cost of player i is greater than or equal to the market price and player i would be better off not producing anything. In other words, we should look for a Nash equilibrium with a smaller number n′ < n of active players. This is only possible if ρ satisfies the stricter inequality ρ < n′ . In the second case, the unit cost of player i is less than the market price and player i would therefore be better off participating by producing some qi∗ > 0. In other words, we should look for a Nash equilibrium with a larger number n′′ > n of active players. In the third case, player i will want to deviate from the candidate equilibrium, and it is not clear where we should look for an alternative equilibrium. This third possibility can be eliminated by a hypothesis on P , namely that ρ < 2. Lemma 2.4. Suppose that ρ < 2 and Q−i ≥ 0. Then g(qi) := π(qi , Q−i , ai ) has a unique global maximum, which is attained in [0, η). Proof. As P (0+ ) may not be finite, the details of the proof depend on whether Q−i > 0 or Q−i = 0. Suppose first that Q−i > 0. Then g(qi ) = qi (P (Q−i + qi ) − ai ) (qi ≥ 0) is twice continuously differentiable everywhere, and in particular at qi = 0. Moreover, for qi ≥ 0, g ′′(qi ) = P ′′ (Q−i + qi ) qi + 2 P ′(Q−i + qi ) qi = 2− ρ(Q−i + qi ) P ′(Q−i + qi ) Q−i + qi qi ρ P ′ (Q−i + qi ) ≤ 2− Q−i + qi ≤ (2 − ρ) P ′(Q−i + qi ) < 0.
(12)
Hence g has a unique global maximum, which is attained in [0, η) since g(0) = 0 and g(qi) < 0 for qi ≥ η. In the case Q−i = 0, qi (P (qi ) − ai ) if qi > 0, g(qi) = 0 if qi = 0. In particular, g(qi) may be discontinuous at qi = 0. As ai < P (0+ ) by assumption, P −1 (ai ) ∈ (0, η] exists and is unique, and g > 0 on (0, P −1(ai )). Consequently, g(0+ ) ≥ 0. We also have g ′(qi ) = qi P ′ (qi ) + P (qi ) − ai , so that g ′ (0+ ) = f1 (0+ ) − ai . It follows from the calculations in the proof of Lemma 2.3 that g ′(0+ ) > 0. Finally, g ′′ (qi ) = (2 − ρ(qi )) P ′(qi ) ≤ (2 − ρ) P ′(qi ) < 0 for all qi > 0. Thus since g < 0 for q > P −1 (ai ), g has a unique global maximum, which is attained in (0, P −1(ai )) ⊂ [0, η).
6
The assumption that ρ < 2 does more than simply eliminate the possibility of competing local maxima: it allows us to implement the approach to characterizing equilibria sketched above. Starting from the one-player equilibrium with player one, who has the lowest cost, we look at whether player two, who has the second lowest cost, wants to participate, in other words if his cost is less than the one-player market price: a2 < P (Q∗1 ). If so, we ask if both the first two players want to participate in the two-player equilibrium, and so on. The following lemma establishes the crucial step. Lemma 2.5. Suppose for some n < N, we have n- and (n + 1)-player candidate equilibria with aggregate production quantities Q∗n and Q∗n+1 respectively, and the individual production levels given by (11) with the appropriate Q∗ . Then player n+1 will want to participate in the n-player equilibrium if and only if he wants to participate in the (n + 1)-player equilibrium. Proof. From (11), player i participates in an n-player candidate equilibrium if and only if ai < P (Q∗n ). Recall from Lemma 2.3 that each Q∗n ∈ (0, η) satisfies fn (Q∗n ) = An , where the functions fn (Q) were defined in (9), and are decreasing on (0, η). For 1 ≤ n < N, it is straightforward to see that fn+1 (Q∗n ) = An − P (Q∗n ), and therefore fn+1 (Q∗n+1 ) − fn+1 (Q∗n ) = an+1 − P (Q∗n ).
(13)
Similarly, fn (Q∗n+1 ) = An+1 − P (Q∗n+1 ), so that fn (Q∗n+1 ) − fn (Q∗n ) = an+1 − P (Q∗n+1 ).
(14)
Then an+1 < P (Q∗n )
⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒
fn+1 (Q∗n+1 ) Q∗n+1 fn (Q∗n+1 ) an+1
< > <
1, it is sufficient just to keep adding the players with equal costs one-by-one in any order. It is clear that if player n + 1 wishes to enter, then so will players n + 2 through n + k, but it suffices to carry out the argument in unit increments. Proposition 2.6 also shows that as players 7
wish to join candidate equilibria, the sequence {Q∗n | 1 ≤ n ≤ N} is first strictly increasing: an+1 < P (Q∗n ) ⇐⇒ Q∗n+1 > Q∗n . It may then become constant: Q∗n1 = Q∗n1 +1 = ... = Q∗n2 for some 1 ≤ n1 < n2 ≤ N if and only if (i) player n1 + 1 is exactly indifferent between entering and remaining out of the n1 -player candidate equilibrium; and (ii) an1 +1 = ... = an2 . Finally, if there is an n′ < N where an′ +1 ≥ P (Q∗n′ ) so that the remaining players after n′ are costed out of participating in any Nash equilibrium, then Q∗n′ +1 ≥ Q∗n′ , and the sequence (Q∗n ) is non-increasing thereafter, and strictly decreasing once some an+1 > P (Q∗n ). When a Nash equilibrium exists and is unique, we denote the equilibrium production of player i as a function of the vector of costs a = (a1 , a2 , ..., aN ) by qi∗ (a), and the equilibrium profit of player i by Gi (a) = π(qi∗ (a), Q∗−i (a), ai ), (15) P where Q∗−i (a) = j6=i qj∗ (a). The functions qi∗ and Gi are essential building blocks in the system of PDEs in Section 3. We have the following corollary. Corollary 2.7. Suppose that ρ < 2. Then the unique Nash equilibrium can be constructed ¯ ∗ = max {Q∗ | 1 ≤ n ≤ N}. Then the unique Nash equilibrium quantities as follows. Let Q n are given by ) ( ¯ ∗ − ai P Q qi∗ (a) = max ¯ ∗ , 0 , 1 ≤ i ≤ N, −P ′ Q and the corresponding profits are
¯ ∗ ) − ai ), Gi (a) = qi∗ (a)(P (Q
1 ≤ i ≤ N.
In particular, qi∗ and Gi are Lipschitz continuous, and the number of active players in the ¯∗ . unique equilibrium is m = min n | Q∗n = Q
Lipschitz continuity follows from the fact that qi∗ and Gi are constructed from compositions of C 1 functions and max operations. Notice that kinks occur in the qi∗ and Gi only when ¯ ∗ for some j, that is, when player j is exactly indifferent between participating aj = P Q or not. Or, to put the same point another way, the qi∗ and Gi are as smooth as P ′ on any region of unit-cost space [0, P (0+ ))N on which the set {i | qi∗ (a) > 0} of active players is constant.
2.2
Examples: Constant Prudence Price Curves
In this section we present formulae for Nash equilibria under a tractable family of price functions for which the relative prudence ρ, defined in (4), is constant. In this case, P satisfies the second-order ordinary differential equation Q P ′′ + ρP ′ = 0. The most natural choices for the two constants of integration are the saturation point η > 0,and the slope at Q the saturation point, −ζ < 0. With these choices, we have P (Q) = −ζηC η , where 1−ρ −1 z if ρ 6= 1, C(z) = 1−ρ log z if ρ = 1, 8
is the canonical solution with η = ζ = 1. However, the translate the units in which P is measured into the units ζ = 1. This leaves us with 1−ρ η 1 − Qη 1−ρ P (Q) = η(log η − log Q)
only role for the constant ζ is to in which a is measured, so we set
ρ 6= 1,
(16)
ρ = 1.
For ρ < 1, the choke price P (0+ ) = η/(1 − ρ) is finite. For ρ ≥ 1, the choke price is infinite. On (0, η], the pricing curve is convex for ρ > 0, affine for ρ = 0, and concave for ρ < 0. As in the general case, we work with the functions fn (Q) = QP ′ (Q) + nP (Q), but now we have much more precise information about these functions. We define nρ = max(1, ⌊ρ⌋), where ⌊ρ⌋ denotes the largest integer less than or equal to ρ. Lemma 2.8. There is no Nash equilibrium for n < ⌊ρ⌋. For n ≥ nρ , there is a unique solution Q∗n ∈ (0, η) to fn (Q) = An , for all 0 ≤ An < nP (0+ ). Proof. For the specific functional forms arising when ρ is constant, we have fn (Q) = (n + 1 − ρ) P (Q) − η. Hence, when ρ > 1 and n ≤ ρ − 1 < ⌊ρ⌋, then fn < 0 on (0, η), and there is no solution Q∗n to the equation fn (Q) = An for An ≥ 0. When n ≥ nρ , fn is decreasing with fn (η) = −η. If P (0+ ) < ∞, then fn (0+ ) ≥ nP (0+ ) and otherwise fn (0+ ) = ∞. Therefore, there is a unique solution, lying in (0, η), to fn (Q) = An , for 0 ≤ An < nP (0+ ). We can then prove existence by starting from the nρ -player candidate equilibrium and adding players until either (i) no further players wish to enter; or (ii) there are no further players. Of course, we must assume that ρ < N + 1, for otherwise there is no n ≤ N for which an n-player candidate equilibrium exists. The only real obstacle to this program is the possibility that, in an n -player candidate equilibrium, the production level qi of player i is not the global maximum of π(·, Q−i, ai ). Since the case ρ < 2 is already covered by Lemma 2.4, we can restrict attention to the case in which ρ ≥ 2. Moreover, if Q−i were to vanish for some i in an n-player candidate equilibrium that was feasible in all other respects, then that candidate equilibrium would also be a oneplayer candidate equilibrium; and if ρ ≥ 2 then there are no one-player candidate equilibria. We can therefore further restrict attention to the case Q−i > 0. Lemma 2.9. Suppose that 2 ≤ ρ < N + 1, and Q−i > 0. Then g(qi) := π(qi , Q−i , ai ) has a unique global maximum, which is attained in [0, η). Proof. Now g(0) = 0 and g(qi ) < 0 for all qi ∈ [η, ∞), so g has a global maximum attained in [0, η]. From (12), we have qi ′′ g (qi ) = 2 − ρ P ′(Q−i + qi ) Q−i + qi 9
for all qi ∈ [0, ∞). There are then two cases to consider. First, if ρ = 2, then g ′′ < 0 everywhere on [0, so g has a global maximum attained in [0, η). Second, if ρ > 2, h ∞), and 2 Q Q−i 2 Q , ∞ and g is then g ′′ < 0 on 0, ρ−2−i and g ′′ > 0 on ρ−2−i , ∞ . Since g ′′ > 0 on 2ρ−2 h Q−i bounded above, we must have g ′ < 0 on 2ρ−2 , ∞ , and uniqueness follows.
Proposition 2.10. Suppose that ρ < N + 1. Then there is a unique Nash equilibrium given as follows: ¯ ρ Q ∗ qi (a) = max P¯ − ai , 0 , 1 ≤ i ≤ N, η
where
P¯ = min {Pn | nρ ≤ n ≤ N} , and ¯= Q
Pn =
An + η , n+1−ρ
nρ ≤ n ≤ N,
(17)
1 P¯ 1−ρ η 1 − (1 − ρ) if ρ 6= 1, η η exp − P¯ η
if ρ = 1.
The corresponding profits are Gi (a) = qi∗ (a)(P¯ − ai ). In particular, qi∗ and Gi are Lipschitz continuous, and the number of active players in the unique equilibrium is m = min n | nρ ≤ n ≤ N, Pn = P¯ .
Proof. For the constant prudence price curves, the formulas are best expressed in terms of the equilibrium prices. By direct calculation, for each nρ ≤ n ≤ N, the unique solution to fn (Q) = An is given by An + η P (Q∗n ) = =: Pn , (18) n+1−ρ since fn (Q) = (n + 1 − ρ)P (Q) − η. Computing P −1 gives 1 η 1 − (1 − ρ) Pn 1−ρ if ρ 6= 1, η Q∗n = η exp − Pn if ρ = 1. η The candidate n-player equilibria are qi∗
Pn − ai = = −P ′ (Q∗n )
Q∗n η
ρ
(Pn − ai ).
We prove existence and uniqueness as in Proposition 2.6. The only difference is that we now confine attention to nρ ≤ n ≤ N. We start with the nρ -player candidate equilibrium. If nρ = 1, then it is obvious player one will wish to make profit and not leave. If nρ > 1, then 0 < nρ + 1 − ρ ≤ 1 and the candidate price Pnρ is guaranteed to exceed the cost of player nρ : Pnρ =
Anρ + η ≥ Anρ + η > anρ . nρ + 1 − ρ 10
That is, player nρ (and therefore all players i < nρ ) will not want to leave the nρ -player candidate equilibrium. Then Lemma 2.5, Proposition 2.6 and Corollary 2.7 hold as in the general case, except we no longer need the restriction ρ¯ < 2 to guarantee unique global maxima. It remains to characterize the transition point where players stop entering the game. As in Corollary 2.7, this occurs when the equilibrium aggregate production level Q∗n first becomes non-increasing, or we reach the maximum number of players N. This is equivalent to when the candidate prices Pn first become non-decreasing or we reach N, and hence the Nash equilibrium price is given by P¯ . ¯ for some As in the general case, kinks occur in the qi∗ and Gi only when aj = P Q j, that is when player j is exactly indifferent between producing or not. Moreover the qi∗ and Gi are as smooth as P ′ on any region of unit-cost space [0, P (0+))N on which the set {i | qi∗ (a) > 0} of active players is constant. Remark 2.11. We note that the general formula for Pn (obtained without setting ζ = 1) is (An +ζ η)/(n+1−ρ), which resolves the seemingly inconsistent dimensions in the numerator of (17). Remark 2.12. There is an even more explicit characterization of m. Consider the function h given by the formula h(n) = η + An−1 − (n − ρ) an . The requirement that a1 < P (0+ ) ensures that h(1) = η − (1 − ρ) a1 > 0. Moreover h(n + 1) − h(n) = (ρ − n) (an+1 − an ). Hence h(n + 1) − h(n) ≥ 0 if and only if n ≤ ρ. Then m is the largest n ∈ {1, 2, ..., N} such that h(n) > 0. In the numerical solutions in Section 5.3, we shall use the linear price function corresponding to ρ = 0, that is, P (Q) = η − Q. In this case, the market price is An + η ¯ P = min |1≤n≤N ; (19) n+1 the quantity produced by player i is qi∗ (a) = max P¯ − ai , 0 for all 1 ≤ i ≤ N; and the 2 profit of player i is Gi (a) = qi∗ (a)(P¯ − ai ) = max P¯ − ai , 0 . Alternatively, in terms of the number of active players m = min{1 ≤ n ≤ N | Pn = P¯ }, we have ! m X 1 aj , Gi (a) = (qi∗ (a))2 , qi∗ (a) = η − mai + m+1 j=1,j6=i for 1 ≤ i ≤ m, and qi∗ = Gi = 0 for i > m. Finally, we mention that ordering of the players by costs is, of course, not crucial to defining the functions qi∗ and Gi in this section. Given a general costs vector, the constructions above are simply modified to first temporarily relabel the firms according to their costs, compute the Nash equilibrium as above, and then return the equilibrium quantities and profits in the original labelling order.
11
3
Differential Game & Exhaustibility
We now introduce the dynamic Cournot game under exhaustibility constraints. Each player i has reserves of a traditional and cheap-to-produce resource (for example oil, by extraction), denoted by xi (t) at time t ≥ 0. We take for simplicity the cost of production from this source to be zero, but reserves are finite (exhaustible). There is also an alternative source that is inexhaustible, but expensive-to-produce (solar power in the energy example), with constant unit cost of production c ∈ [0, P (0+ )). Player i chooses a dynamic production rate q¯i that is a Markov strategy:1 q¯i = q¯i (x(t)), where x(t) = (x1 (t), · · · , xN (t)). As long as xi > 0, player i has the choice between producing from the cheap or expensive sources. After xi hits zero, he can only produce from the costlier alternative, which never runs out. We shall suppose, at first, that no player produces from the more expensive source as long as the cheaper one is available,2 and we will discuss how this could be validated a posteriori. Therefore, reserves of his traditional resource deplete according to dxi = −¯ qi (x(t)), xi > 0. dt (To lighten the notation, we do not denote the dependence of x on the q¯i .) The market price is governed by a Cournot competition with the price function P as before, satisfying Assumption 2.2. We assume that, given a cost vector a satisfying (3), there is a unique Nash equilibrium q ∗ (a) of the static Cournot game. Some general conditions for this were given in Proposition 2.6, and, for a specific family of price functions, in Proposition 2.10.
3.1
Dynamic Cournot Competition
Given initial reserves xi (0) ≥ 0, player i wants to maximize his discounted lifetime profit Z ∞ ¯ −i (x(t)), c1I{x (t)=0} dt, e−rt π q¯i (x(t)), Q i 0
where 1I denotes the indicator P function, r > 0 is a discount rate, the profit function π was ¯ defined in (1), and Q−i = j6=i q¯j . Note that the cost of production rises from zero to c when reserves xi run out, as denoted in the third argument of π in the integral. ∗ We look for a Markov Perfect Nash equilibrium q¯ ∗ (x(t)) = (¯ q1∗ (x(t)), · · · , q¯N (x(t))) such ∗ that, for each player i, and each initial state x(0), q¯i is the best response whenPall the other ¯∗ = players play their equilibrium strategies. Therefore, with the notation Q ¯j∗ , −i j6=i q Z
∞
−rt
e
0
π
¯ ∗−i (x(t)), c1I{x (t)=0} q¯i∗ (x(t)), Q i
dt ≥
Z
0
∞
¯ ∗−i (x(t)), c1I{x (t)=0} dt, e−rt π q¯i (x(t)), Q i
N for any Markov strategy q¯i of player i, and for all x(0) ∈ RN + = [0, ∞) . The requirement that the equilibrium strategies are independent of the initial resource levels x(0) is equivalent, 1
Markov strategies are also sometimes called feedback or closed-loop strategies. According to the lead editorial in The Times of London on 13 July, 2009: “No sane energy company would, while fossil fuels are still plentiful, voluntarily opt for a more expensive, less reliable energy source.” 2
12
in our setting, to the requirement that the equilibrium is perfect, sometimes called subgame perfect. This excludes equilibria with so-called “incredible threats” whereby players may make extreme, but unrealistic, threats of increased production if another player deviates from a certain path. We refer to [14] and the textbooks [9, 10] for further discussion and references on this issue. We give an informal motivation for the dynamic programming PDEs we shall use to construct Nash equilibria for these problems. First, consider any continuous Markov perfect strategy {¯ qj (x(t)) | t ≥ 0, 1 ≤ j ≤ N}, and the profits starting at time s ≥ 0: Z ∞ q¯ ¯ −i (x(t)), c1I{x (t)=0} dt, 1 ≤ i ≤ N. vi (x(s)) = e−r(t−s) π q¯i (x(t)), Q (20) i s
Let x = x(0) denote any interior point (xj > 0, for all 1 ≤ j ≤ n), so all players start with some intitial reserves.3 Then, differentiating (20) with respect to s and setting s = 0 gives the partial differential equation ¯ −i (x), 0) − π(¯ qi (x), Q
N X
∂viq¯ q¯j (x) − rviq¯ = 0, ∂xj j=1
which can be re-written as X ∂viq¯ ∂viq¯ ¯ q¯j (x) − − rviq¯ = 0. π q¯i (x), Q−i (x), ∂xj ∂xj j6=i
(21)
(22)
Then Bellman-principle arguments4 reduce each player’s optimization problem to a local optimization, and the search for a Markov perfect equilibrium to a search for a local static Nash equilibrium, which, in this case, amounts to optimizing π in (22) with respect to its first argument, ∗with the second argument fixed at the other players’ equilibrium strategies. Writing vi = viq¯ for the value functions using the equilibrium policies: Z ∞ ¯ ∗ (x(t)), c1I{x (t)=0} dt, vi (x) = e−rt π q¯i∗ (x(t)), Q −i i 0
we have
X ∂vi ∂v i ∗ ¯ (x), − q¯j∗ (x) max π qi , Q − rvi = 0, −i qi ≥0 ∂xi ∂xj j6=i
i = 1, · · · , N.
(23)
where x ∈ RN + with xj > 0 for 1 ≤ j ≤ N, that is when all players have some reserve of the traditional resource. We assume throughout that each vi is continuously differentiable up to the axes: vi ∈ ∂vi C 1 (RN ¯i∗ (x) is continuous at all x ∈ RN + ); and that q + . We observe from (23) that ∂xi enters as a “shadow cost” for player i at the differential level. The interpretation as a cost is ∂vi legitimate as we naturally expect ∂x ≥ 0: adding more reserves increases the value function. i 3 4
The boundary cases when some xj = 0 are dealt with in Section 3.2. See, for instance, [15], [8, Section 8.2], [1, Section 6.5.2], or [6, Section 4.2].
13
Comparison with (2) reveals the differential Nash equilibrium problem in the PDEs is just the one-period game with the role of the costs ai played by the partial derivatives ∂vi /∂xi . ∗ For a fixed x ∈ RN + , if there is a unique Nash equilibrium q (a) for the static game with ai =
∂vi (x), ∂xi
i = 1, · · · , N,
then we re-write equations (23) as X ∂vi − rvi = 0, Gi (Dv) − qj∗ (Dv) ∂xj j6=i
where we define
i = 1, · · · , N,
(24)
∂v1 ∂vN Dv = diag(∇v) = ,··· , , ∂x1 ∂xN and recall that Gi (a) = qi∗ (a)(P (Q∗ ) − ai ) is equilibrium profit function of the static game. The equilibrium production rates of the exhaustible resource at time t are given by q¯i∗ (x(t)) = qi∗ (Dv(x(t))). Note that the definition of Nash equilibrium and the constructions of qi∗ (a) and Gi (a) in Propositions 2.6 and 2.10 take care of the fact that not all players may participate at all resource levels x, depending on the vector of shadow costs a = Dv(x) at that point. However they encompass the fact that there are always potentially N active players. For the majority of the paper, we shall treat the cases where all players participate, but we shall discuss the situation where one player may be blockaded in the two-player dynamic game in Section 5. At this level of generality, we are not able to provide reasonable conditions for existence and uniqueness of a solution to the system (24), equipped with appropriate boundary conditions discussed in the following section, let alone solutions with sufficient regularity to generate a unique Nash equilibrium with well-behaved strategies. We will proceed by staying relatively close to a case which is well-understood. In the next section, we address the issue of exhaustibility and boundary conditions.
3.2
Exhaustibility
When a player has exhausted his reserves of the cheap resource, he can turn to the alternative means of production which, while more costly than the original one, allows the exhausted player to remain in the game, but in a disadvantaged position. In the energy example, there are alternative “backstop” technologies, such as solar power or steam-extracted oil shales, that an energy supplier may resort to when his reserves of oil run out, both of which are more expensive than delivering energy by extracting oil. We consider the case xi = 0, when player i has exhausted his supply. Then we have dxi = 0. dt The HJ equation for vi (x1 , · · · , xi−1 , 0, xi+1 , · · · , xN ) becomes X ∗ ∂vi ¯ ∗ (x), c − qj (Dv) − rvi = 0, max π qi , Q −i qi ≥0 ∂xj j6=i 14
or Gi (D−i v) − where we define D−i v =
X
qj∗ (D−i v)
j6=i
∂vi − rvi = 0, ∂xj
∂vi−1 ∂vi+1 ∂vN ∂v1 ,··· , , c, ,··· , ∂x1 ∂xi−1 ∂xi+1 ∂xN
(25)
.
Similarly, the HJ equation for vj (x1 , · · · , xi−1 , 0, xi+1 , · · · , xN ), j 6= i becomes Gj (D−i v) −
X
qk∗ (D−i v)
k ∈{i,j} /
∂vj − rvj = 0. ∂xk
(26)
These are then a system of N equations on the hyperplane xi = 0 involving only partial derivatives along the plane. We can proceed similarly to the cases in which more and more players have exhausted their reserves until we reach the fully exhausted case. Here, all the players are using the inexhaustible alternative resource, and so produce at the constant rate qi∗ (c1), where 1 denotes the N-vector of ones. It follows that the value functions are just the constant Gi (c1) . (27) vi (0) = r This serves as the initial condition for the one-player ODEs on the lines. Once solved, these are axis Dirichlet boundary conditions for the two-player PDEs on the planes, and so on. Now if the value functions are found to satisfy ∂vi < c in {xi > 0}, ∂xi then our initial hypothesis, that no producer will use the alternative technology while the cheaper one is available, will be validated. This is clear because if we introduced additional control variables in the interior so that each player could choose both a quantity from reserves and a quantity from the alternative source, then in each game at the differential level, none ∂vi would be produced from the alternative as long as the shadow cost ∂x is smaller than c. This i is indeed the case for the approximate solutions in Section 4.2, and the numerical solutions of Section 5.3, and we shall assume it to be the case for the remainder of the paper.
3.3
Neumann Boundary Conditions
When all players participate at all resource levels, it is possible to replace the Dirichlet boundary conditions coming from (26) by simpler Neumann boundary conditions. We reiterate that the Dirichlet conditions for the value functions are found by solving the chain of games starting with the case of all players exhausted, then the game where one has reserves and N − 1 players are using the alternative technology, up to N − 1 with reserves and one using the alternative. Recall that we have assumed that we have continuity of the first derivatives of the value functions up to the boundaries and continuity of the equilibrium policies q¯i∗ (x) at all x ∈ RN +.
15
ˆ denote the projection of x Fix 1 ≤ i ≤ N and a point x ∈ RN + with all xj > 0, and let x onto {xi = 0}. For player i, his equilibrium production on xi = 0 is given by ∂vi−1 ∂vi+1 ∂vN ∂v1 ∗ ∗ ∗ (ˆ x), · · · , (ˆ x), c, (ˆ x), · · · , (ˆ x) , q¯i (ˆ x) = qi (D−i v(ˆ x)) = qi ∂x1 ∂xi−1 ∂xi+1 ∂xN whereas in the interior it is q¯i∗ (x)
=
qi∗ (Dv(x))
=
qi∗
∂v1 ∂vN (x), · · · , (x) . ∂x1 ∂xN
ˆ . In general, we expect that From continuity of q¯i∗ (x), qi∗ (Dv(x)) → qi∗ (D−i v(ˆ x)) as x → x ∗ qi (a) is a continuous function, which was true in the cases of Corollary 2.7 and Proposition 2.10. Further, we expect the static equilibrium quantity qi∗ (a) to be decreasing in its i-th argument, as long as qi∗ > 0. Specifically, if it is the case that ∂qi∗ (D−i v(ˆ x)) < 0 and qi∗ (D−i v(ˆ x)) > 0, ∂ai then we can conclude that
∂vi = c on xi = 0. (28) ∂xi The interpretation of this expression is that, on hitting the boundary xi = 0, the shadow cost of player i turns into the real cost c. As long as player i participates in the game on xi = 0, then we have the Neumann boundary condition (28). Comparison of (26) with (24), which we re-write as Gj (Dv) −
X
qk∗ (Dv)
k6=j
∂vj − rvj = 0, ∂xk
yields
∂vj = 0 on xi = 0, j 6= i (29) ∂xi provided qi∗ (D−iv) 6= 0. Therefore, as long as player i still participates in the game on xi = 0, when he is forced to use the alternative technology, the shadow cost of the other players is zero. Indeed, when the cost c is small enough, all players will participate at all resource levels. For example, in the case of the constant prudence price curves of Section 2.2, we need only consider the extreme case where there are N − 1 producers with the minimum unit cost of zero, and one with the maximum possible unit cost c. If ρ ∈ [N, N + 1), then only a full N-player Nash equilibrium is possible. As the candidate market price from (17) is PN =
η+c > c, N +1−ρ
since 0 < N + 1 − ρ ≤ 1, all players participate in this case for any c < ∞. In the case ρ < N, the candidate market price PN −1 = η/(N − ρ) exceeds c if c
0. We remark that the inexhaustible limit (31) is also the behaviour for large discounting rate r or large resources. Indeed if we write the value functions as vi (x; r) to stress the discount rate, it is easy to check from the PDEs (24) and boundary conditions (28) and (29), that 1 vi (x; r) = vi (rx; 1), r so formal asymptotics in the limits of large (or small) r or ||x|| are analogous calculations.
4
Small Exhaustibility Approximation
When c is small, we are close to the inexhaustible game played repeatedly, and we may expect that all players participate at all resource levels. In preparation for an approximation in this case for the dynamic game, we first analyze the effect of small-costs on the static game.
4.1
Static Game Small Cost Perturbation
We return to the static game, whose Nash optimal strategies are given by qi∗ (a), and equilibrium profits are Gi (a) = qi∗ (a)(P (Q∗ ) − ai ), (32) where Q∗ satisfies (7). We assume costs a are such that all players participate in the equilibrium and the qi∗ (a) (and hence the Gi (a)) are differentiable. A small costs Taylor expansion gives X Gi (a) ≈ Gi (0) + Aai + B aj , j6=i
where we define the constants
A=
∂Gi (0), ∂ai
B=
17
∂Gi (0), ∂aj
j 6= i,
(33)
which are independent of i and j. Similarly, the strategies are given approximately by X aj , (34) qi∗ (a) ≈ γ + λai + µ j6=i
where we define γ=
∂qi∗ λ= (0), ∂ai
qi∗ (0),
∂qi∗ µ= (0), ∂aj
j 6= i,
(35)
which are again independent of i and j. Further, we define the constant of relative prudence at the zero cost equilibrium solution by ρ0 = −Nγ
P ′′ (Nγ) , P ′(Nγ)
(36)
which is just ρ(Nγ), where ρ(q) was defined in (4). Then we have the following expressions for (A, B, λ, µ) in terms of (γ, ρ0 , P ′(Nγ)). Proposition 4.1. We assume ρ0 < (N + 1). The perturbation coefficients (A, B, λ, µ) can be expressed as 2N − (2 − N −1 )ρ0 A = −γ , (37) (N + 1) − ρ0 2 − N −1 ρ0 B = γ , (38) (N + 1) − ρ0 N − (1 − N −1 )ρ0 1 , (39) λ = P ′ (Nγ) (N + 1) − ρ0 1 − N −1 ρ0 1 . (40) µ = − ′ P (Nγ) (N + 1) − ρ0 Proof. Let θ = λ + (N − 1)µ. Differentiating the summed first-order conditions (7) with respect to ai and setting a = 0 gives (N + 1)θP ′ (Nγ) − 1 + NγθP ′′ (Nγ) = 0, which gives θ=
1 1 = . (N + 1)P ′ (Nγ) + NγP ′′ (Nγ) P ′ (Nγ)((N + 1) − ρ0 )
Differentiating (6) with respect to ai and setting a = 0 gives θP ′ (Nγ) − 1 + λP ′ (Nγ) + γθP ′′ (Nγ) = 0, which yields λ=
1 P ′ (Nγ)
(1 − θ(P ′ (Nγ) + γP ′′ (Nγ))).
Re-arranging leads to (39) and (40). 18
Next, differentiating (32) with respect to ai and setting a = 0 gives
But, from (7) with AN =
P
A = λP (Nγ) + γ(θP ′(Nγ) − 1).
j
aj = 0, we have P (Nγ) = −γP ′ (Nγ). Therefore,
A = γ(θ − λ)P ′ (Nγ) − γ = γ(N − 1)µP ′ (Nγ) − γ, and (37) follows after substituting from (40). Similarly, differentiating (32) with respect to aj (j 6= i) and setting a = 0 gives B = γ(θ − µ)P ′(Nγ), and (38) follows after substitution for θ and µ. We comment that since ρ0 < (N + 1) (and N ≥ 2), the formulas (38-40) imply that B > 0,
and λ < 0,
(41)
while (37) implies that A ≤ 0 for ρ0 ≤ and A > 0 otherwise. Formula (40) yields
2N 2 , 2N − 1
(42)
µ ≥ 0 for ρ0 ≤ N, and µ < 0 otherwise. In other words, player i’s profits increase when any other player’s cost aj is increased from zero, and he also increases his production for ρ0 ≤ N. When his own cost ai is increased from zero, he decreases his production, but his profit may increase or decrease depending on ρ0 . Finally, if costs for all the players are increased by the same amount, ai = c, ∀i, then Gi (c1) ≈ Gi (0) + (A + (N − 1)B)c, so each player’s equilibrium profit increases with cost according to the sign of A + (N − 1)B. From (37) and (38), we have A + (N − 1)B = γ
(ρ0 − 2) , (N + 1) − ρ0
so profits actually increase with costs for ρ0 > 2, suggesting that sufficient curvature induces a degree of mutually beneficial production. Similarly, the optimal production quantities under a symmetric increase in costs are approximated as qi∗ (c1) ≈ qi∗ (0) + (λ + (N − 1)µ)c. From (39) and (40), λ + (N − 1)µ =
1 P ′ (Nγ)((N
+ 1) − ρ0 )
< 0,
(43)
so each player’s equilibrium production decreases with an across-the-board cost increase. 19
4.2
Differential Game Small Cost Perturbation
We look for an approximate solution of (24), with boundary conditions (28)-(29), of the form vi (x) =
Gi (0) (1) + cvi (x) + o(c), r
(1)
for some functions vi to be found. Inserting the expansion into the PDEs and boundary conditions leads to the linearized system (1)
(1)
A
(1)
X ∂vj X ∂v ∂vi (1) i +B −γ − rvi = 0, ∂xi ∂xj ∂xj j6=i
(44)
j6=i
(1)
∂vi |x =0 = 1, ∂xi i (1) ∂vi |x =0 = 0 (j 6= i), ∂xj i
(45) (46)
where (A, B, γ) were defined in (33) and (35). We have the following explicit solution. Proposition 4.2. Assume that ρ0
Sam Howison†
Ronnie Sircar‡
August 14, 2009
Abstract We study N -player continuous-time Cournot games in an oligopoly where firms choose production quantities. These are nonzero-sum differential games, whose value functions may be characterized by systems of nonlinear Hamilton-Jacobi partial differential equations. When resources are in finite supply, such as oil, exhaustibility enters as boundary conditions for the PDEs. We analyze the problem when there is an alternative, but expensive, technology (for example solar power for energy production), and give an asymptotic approximation in the limit of small exhaustibility. We illustrate the two-player problem by numerical solutions, and discuss the impact of limited oil reserves on production and oil prices in the duopoly case.
Contents 1 Introduction
2
2 Static Cournot Game 2.1 General Price Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Examples: Constant Prudence Price Curves . . . . . . . . . . . . . . . . . .
3 4 8
3 Differential Game & Exhaustibility 3.1 Dynamic Cournot Competition . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Exhaustibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Neumann Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . .
12 12 14 15
4 Small Exhaustibility Approximation 4.1 Static Game Small Cost Perturbation . . . . 4.2 Differential Game Small Cost Perturbation . 4.3 Time to Exhaustion in the Symmetric Game 4.4 Illustration: Duopoly . . . . . . . . . . . . .
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Faculty of Economics, Cambridge University, Sidgwick Avenue, Cambridge CB3 9DD; [email protected] † OCIAM, Mathematical Institute, Oxford University, 24-29 St. Giles’, Oxford OX1 3LB; [email protected] ‡ ORFE Department, Princeton University, Sherrerd Hall, Princeton NJ 08544; [email protected]. Work partially supported by NSF grant DMS-0807440.
1
5 Duopoly Extraction Problem with 5.1 The Axis Game & Blockading . . 5.2 Type of the PDEs . . . . . . . . . 5.3 Numerical Solutions . . . . . . . .
the Linear . . . . . . . . . . . . . . . . . . . . .
6 Conclusions
1
Pricing . . . . . . . . . . . . . . .
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24 25 30 33 37
Introduction
The problem of dwindling oil reserves and its impact on energy supply and prices is of longstanding importance. One way to analyze the issues is to view energy markets as being governed by a small number of competitive firms or countries, that is as oligopolies, and to model the formation of prices and supplies within this competitive framework. Game theory provides a natural way to frame the outcome of competition within different choices of market mechanism. Exhaustibility, meanwhile, requires analysis of how anticipation of changing resources impacts production and prices, and therefore leads to dynamic games. Here we study continuous-time (or differential) games arising from competition over a resource in limited supply. The games are nonzero-sum as all players act to maximize their own profits. Typical models of industrial organization in the economics literature are restricted to the study of one or two-period games. For ordinary and stochastic zero-sum differential games, there is a fairly general theory [7, 11], including a viscosity theory for their associated (scalar) Hamilton-Jacobi-Bellman-Isaacs PDEs. In the nonzero-sum case where systems of equations for the value functions of all the players arise, there is, to our knowledge, no similar general theory. Some recent books that discuss nonzero-sum deterministic and stochastic differential games are [1, 6]. There is also a literature on exhaustibility (or capacity constraints), but not, primarily, in the context of continuous-time models. We mention [5] as a reference for the literature on exhaustible resources up through the 1970s. When the quantity being produced is in finite supply, such as oil, exhaustibility is a “game-changer”, and enters as boundary conditions for the PDEs. We analyze the problem when there is an alternative resource (for example solar technology for energy production), which is inexhaustible, but much more costly to produce than extracting oil. Of interest is the impact on oil extraction rates, and hence market prices, as reserves run out and energy production must switch to more expensive renewable sources. We begin by analysing the static N-player Cournot game as a function of the costs of production of the firms. When we move to the dynamic problem in Section 3, exhaustibility acts like a varying cost, that depends on the dynamic game’s value functions, in a static game at the infinitesimal level. Therefore, we devote some effort in Section 2 to establish existence and uniqueness of a Nash equilibrium for the static game with players who have different costs. The static game equilibrium production and profit functions are essential ingredients for the partial differential equations characterizing the dynamic game in Section 3. In Section 4, we derive a perturbation approximation for the case when the cost of the alternative technology is small. Section 5 studies in detail and with numerical PDE solutions a specific two-player example. In particular, we discuss the issue of one firm being blockaded out of competition by the dominance of the other. Section 6 concludes.
2
2
Static Cournot Game
In the bulk of the paper we will analyse a dynamic version of the Cournot model of competition: producers of the resource set quantities which they bring to the market, and the price is then determined from the total quantity produced. Before moving to the dynamic game with exhaustibility, we analyze the static (one-period, or stage) game and introduce notation from it that will be needed later. We take as given a price (or inverse-demand) function P : (0, ∞) → R that gives market price (per unit) as a function of quantity produced and put on the market. There are N ≥ 1 players. Each player i chooses a quantity qi ∈ [0, ∞) to produce at unit cost of production ai ≥ 0, and the market price is determined from the total production. In the original example described by Cournot [4], the inexhaustible resource being produced was mineral water. Once each player chooses his quantity, the market price is given by P (Q),
where Q =
N X
qj .
j=1
The profit of player i is the quantity he produces multiplied by price minus cost: qi (P (Q−i + qi ) − ai ) if qi > 0, (1) π(qi , Q−i , ai ) = 0 if qi = 0, P where Q−i = j6=i qj is total production by the players other than i. We allow for the possibility that P (0+ ) = +∞, but specify π(qi , Q−i , ai ) = 0 when qi = 0: if a player does not produce anything, then he makes no profit. Each player seeks to maximize his own profit, taking the quantities produced by the other players as given. More precisely: ∗ Definition 2.1. A Nash equilibrium is a vector q ∗ = (q1∗ , q2∗ , ..., qN ) ∈ [0, ∞)N such that, for all i, π(qi∗ , Q∗−i , ai ) = max π(qi , Q∗−i , ai ), (2) qi ∈[0,∞) P where Q∗−i = j6=i qj∗ . That is, each player’s equilibrium production qi∗ maximizes his own profit π(·, Q∗−i, ai ) when the other N − 1 players produce their equilibrium quantities. If, in addition, qi∗ > 0 for all i, then we call q ∗ an interior Nash equilibrium.
The main aim of this section is to show that, under suitable conditions on the price function P and the cost vector a = (a1 , a2 , · · · , aN ), a Nash equilibrium exists and is unique. Much of the literature on static Cournot games assumes identical costs among the players (so that all or none will participate), or assumes all costs are small enough that there is an interior Nash equilibrium. In the models we study here, costs are linear in quantity (constant marginal cost), but we shall need to consider the situation of hetergeneous costs among the players, which will be related to different proximities to exhaustion in the dynamic game of Section 3. An important issue arising from this is that it may be too costly for some players to participate. To our knowledge, this aspect has not been fully addressed before, even in the static game. We refer to [16, Chapter 4] for a discussion and references on general existence results for static Cournot games. Our primary assumptions on P are: 3
Assumption 2.2. The price function P is twice continuously differentiable, with P ′ < 0 everywhere on (0, ∞); and there exists η ∈ (0, ∞) such that P (η) = 0. The first part of this assumption is natural: the greater total production, the less the market will be willing to pay per unit. The second part implies that P (Q) < 0 for Q > η: if there is over-production, players have to pay to have the surplus removed. Nonetheless, it should be noted that negative prices will play no role in our analysis, because profitmaximizing players with positive costs will never produce at a level at which prices are negative. We shall refer to η as the saturation point. We order the firms by their costs and assume they are strictly less than the choke price P (0+ ): 0 ≤ a1 ≤ a2 ≤ ... ≤ aN < P (0+ ). (3)
When some firms have equal costs, the ordering is arbitrary and does not affect the analysis that follows. The assumption that ai < P (0+ ) for all i ensures that, in any Nash equilibrium P ∗ q ∗ , total production Q∗ = N i=1 qi will be strictly positive, for if all players other than i produce nothing, so that qj = 0 for all j 6= i, then player i can make a strictly positive profit qi (P (qi ) − ai ) by producing a small positive amount qi (and may be able to do much better). Having all players produce nothing is therefore not a Nash equilibrium. The assumption that ai ≥ 0 for all i ensures that P (Q∗ ) > 0. In particular, Q∗ < η. The behaviour of P is best characterized in terms of the relative prudence of P , namely ρ(Q) = −
Q P ′′ (Q) . P ′ (Q)
(4)
Our terminology here is adapted from [12]. We also define ρ = sup ρ(Q).
(5)
Q∈(0,∞)
We turn now to the detailed analysis of Nash equilibrium, first for general price functions P , and then, in Section 2.2, for a convenient family of examples which are amenable to explicit calculations.
2.1
General Price Functions
Suppose that q ∗ is an interior Nash equilibrium. Then, for all i ∈ {1, 2, ..., N}, qi∗ must satisfy the first-order condition 0=
∂π ∗ ∗ (q , Q , ai ) = qi∗ P ′ (Q∗−i + qi∗ ) + P (Q∗−i + qi∗ ) − ai . ∂qi i −i
(6)
Summing over i, we obtain 0 = Q∗ P ′ (Q∗ ) + N P (Q∗ ) − AN ,
(7)
P PN ∗ where Q∗ = Q∗−i + qi∗ = N i=1 qi is total production and AN = i=1 ai is the sum of the unit ∗ costs. In other words, Q must satisfy the scalar equation fN (Q) = AN , where fN (Q) = QP ′ (Q) + N P (Q), 4
Q > 0.
On the other hand, given a solution of this equation, making qi∗ the subject of equation (6), a candidate Nash equilibrium is P (Q∗ ) − ai . (8) qi∗ = −P ′ (Q∗ ) Using (6), we can also express the candidate equilibrium quantities in (8) as ! ∗ P (Q ) − a i qi∗ = PN Q∗ . ∗ j=1 (P (Q ) − aj )
This has the interpretation that, once the equilibrium total quantity Q∗ is determined, the fraction produced by player i is the deviation of his cost ai from the market price P (Q∗ ) relative to the total deviation of all players’ costs from the price. However, some qi∗ defined by (8) may be negative, and so we must also consider Nash equilibria in which some players do not produce. For 1 ≤ n ≤ N, we define fn (Q) = QP ′ (Q) + n P (Q), and An =
Pn
i=1
Q > 0,
(9)
ai . We then have the following lemma.
Lemma 2.3. Fix n ∈ {1, · · · , N}, and suppose that ρ < n + 1. Then there is a unique Q∗n ∈ (0, η) such that fn (Q∗n ) = An . Proof. It is sufficient to show that fn is decreasing with fn (0+ ) > An and fn (η) < 0, so there is a unique root of fn (Q) = An in (0, η). We compute fn′ (Q) = QP ′′ (Q) + (n + 1) P ′(Q) = (n + 1 − ρ(Q)) P ′ (Q) ≤ (n + 1 − ρ) P ′(Q).
(10)
Hence fn′ < 0 on (0, ∞). We also have fn (η) = ηP ′(η) < 0 (since P (η) = 0). We therefore need only show that fn (0+ ) > An . In the case where P (0+ ) is finite and P ′ (0+ ) exists and is finite, fn (0+ ) = nP (0+ ) > An , from (3). The case where the choke price is infinite (P (0+ ) = +∞) is a little more involved, and is handled in the appendix. Then, for each n > max(0, ρ − 1), we have the following n-player candidate Nash equilibrium: ( P (Q∗n )−ai for 1 ≤ i ≤ n, ∗ −P ′ (Q∗n ) (11) qi = 0 for n + 1 ≤ i ≤ N, where Q∗n is the unique solution of fn (Q) = An , and we recall that the players are ordered by their production costs ai . This candidate equilibrium can fail to be a Nash equilibrium of the game as a whole in one of three ways: 1. it may happen that qi∗ < 0 for some 1 ≤ i ≤ n; 2. it may happen that ai < P (Q∗n ) for some n + 1 ≤ i ≤ N; or 3. it may happen that, for some 1 ≤ i ≤ n, qi∗ is not a global maximum of π(·, Q−i , ai ). 5
The first case occurs if and only if ai > P (Q∗n ), that is, the unit cost of player i is greater than or equal to the market price and player i would be better off not producing anything. In other words, we should look for a Nash equilibrium with a smaller number n′ < n of active players. This is only possible if ρ satisfies the stricter inequality ρ < n′ . In the second case, the unit cost of player i is less than the market price and player i would therefore be better off participating by producing some qi∗ > 0. In other words, we should look for a Nash equilibrium with a larger number n′′ > n of active players. In the third case, player i will want to deviate from the candidate equilibrium, and it is not clear where we should look for an alternative equilibrium. This third possibility can be eliminated by a hypothesis on P , namely that ρ < 2. Lemma 2.4. Suppose that ρ < 2 and Q−i ≥ 0. Then g(qi) := π(qi , Q−i , ai ) has a unique global maximum, which is attained in [0, η). Proof. As P (0+ ) may not be finite, the details of the proof depend on whether Q−i > 0 or Q−i = 0. Suppose first that Q−i > 0. Then g(qi ) = qi (P (Q−i + qi ) − ai ) (qi ≥ 0) is twice continuously differentiable everywhere, and in particular at qi = 0. Moreover, for qi ≥ 0, g ′′(qi ) = P ′′ (Q−i + qi ) qi + 2 P ′(Q−i + qi ) qi = 2− ρ(Q−i + qi ) P ′(Q−i + qi ) Q−i + qi qi ρ P ′ (Q−i + qi ) ≤ 2− Q−i + qi ≤ (2 − ρ) P ′(Q−i + qi ) < 0.
(12)
Hence g has a unique global maximum, which is attained in [0, η) since g(0) = 0 and g(qi) < 0 for qi ≥ η. In the case Q−i = 0, qi (P (qi ) − ai ) if qi > 0, g(qi) = 0 if qi = 0. In particular, g(qi) may be discontinuous at qi = 0. As ai < P (0+ ) by assumption, P −1 (ai ) ∈ (0, η] exists and is unique, and g > 0 on (0, P −1(ai )). Consequently, g(0+ ) ≥ 0. We also have g ′(qi ) = qi P ′ (qi ) + P (qi ) − ai , so that g ′ (0+ ) = f1 (0+ ) − ai . It follows from the calculations in the proof of Lemma 2.3 that g ′(0+ ) > 0. Finally, g ′′ (qi ) = (2 − ρ(qi )) P ′(qi ) ≤ (2 − ρ) P ′(qi ) < 0 for all qi > 0. Thus since g < 0 for q > P −1 (ai ), g has a unique global maximum, which is attained in (0, P −1(ai )) ⊂ [0, η).
6
The assumption that ρ < 2 does more than simply eliminate the possibility of competing local maxima: it allows us to implement the approach to characterizing equilibria sketched above. Starting from the one-player equilibrium with player one, who has the lowest cost, we look at whether player two, who has the second lowest cost, wants to participate, in other words if his cost is less than the one-player market price: a2 < P (Q∗1 ). If so, we ask if both the first two players want to participate in the two-player equilibrium, and so on. The following lemma establishes the crucial step. Lemma 2.5. Suppose for some n < N, we have n- and (n + 1)-player candidate equilibria with aggregate production quantities Q∗n and Q∗n+1 respectively, and the individual production levels given by (11) with the appropriate Q∗ . Then player n+1 will want to participate in the n-player equilibrium if and only if he wants to participate in the (n + 1)-player equilibrium. Proof. From (11), player i participates in an n-player candidate equilibrium if and only if ai < P (Q∗n ). Recall from Lemma 2.3 that each Q∗n ∈ (0, η) satisfies fn (Q∗n ) = An , where the functions fn (Q) were defined in (9), and are decreasing on (0, η). For 1 ≤ n < N, it is straightforward to see that fn+1 (Q∗n ) = An − P (Q∗n ), and therefore fn+1 (Q∗n+1 ) − fn+1 (Q∗n ) = an+1 − P (Q∗n ).
(13)
Similarly, fn (Q∗n+1 ) = An+1 − P (Q∗n+1 ), so that fn (Q∗n+1 ) − fn (Q∗n ) = an+1 − P (Q∗n+1 ).
(14)
Then an+1 < P (Q∗n )
⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒
fn+1 (Q∗n+1 ) Q∗n+1 fn (Q∗n+1 ) an+1
< > <
1, it is sufficient just to keep adding the players with equal costs one-by-one in any order. It is clear that if player n + 1 wishes to enter, then so will players n + 2 through n + k, but it suffices to carry out the argument in unit increments. Proposition 2.6 also shows that as players 7
wish to join candidate equilibria, the sequence {Q∗n | 1 ≤ n ≤ N} is first strictly increasing: an+1 < P (Q∗n ) ⇐⇒ Q∗n+1 > Q∗n . It may then become constant: Q∗n1 = Q∗n1 +1 = ... = Q∗n2 for some 1 ≤ n1 < n2 ≤ N if and only if (i) player n1 + 1 is exactly indifferent between entering and remaining out of the n1 -player candidate equilibrium; and (ii) an1 +1 = ... = an2 . Finally, if there is an n′ < N where an′ +1 ≥ P (Q∗n′ ) so that the remaining players after n′ are costed out of participating in any Nash equilibrium, then Q∗n′ +1 ≥ Q∗n′ , and the sequence (Q∗n ) is non-increasing thereafter, and strictly decreasing once some an+1 > P (Q∗n ). When a Nash equilibrium exists and is unique, we denote the equilibrium production of player i as a function of the vector of costs a = (a1 , a2 , ..., aN ) by qi∗ (a), and the equilibrium profit of player i by Gi (a) = π(qi∗ (a), Q∗−i (a), ai ), (15) P where Q∗−i (a) = j6=i qj∗ (a). The functions qi∗ and Gi are essential building blocks in the system of PDEs in Section 3. We have the following corollary. Corollary 2.7. Suppose that ρ < 2. Then the unique Nash equilibrium can be constructed ¯ ∗ = max {Q∗ | 1 ≤ n ≤ N}. Then the unique Nash equilibrium quantities as follows. Let Q n are given by ) ( ¯ ∗ − ai P Q qi∗ (a) = max ¯ ∗ , 0 , 1 ≤ i ≤ N, −P ′ Q and the corresponding profits are
¯ ∗ ) − ai ), Gi (a) = qi∗ (a)(P (Q
1 ≤ i ≤ N.
In particular, qi∗ and Gi are Lipschitz continuous, and the number of active players in the ¯∗ . unique equilibrium is m = min n | Q∗n = Q
Lipschitz continuity follows from the fact that qi∗ and Gi are constructed from compositions of C 1 functions and max operations. Notice that kinks occur in the qi∗ and Gi only when ¯ ∗ for some j, that is, when player j is exactly indifferent between participating aj = P Q or not. Or, to put the same point another way, the qi∗ and Gi are as smooth as P ′ on any region of unit-cost space [0, P (0+ ))N on which the set {i | qi∗ (a) > 0} of active players is constant.
2.2
Examples: Constant Prudence Price Curves
In this section we present formulae for Nash equilibria under a tractable family of price functions for which the relative prudence ρ, defined in (4), is constant. In this case, P satisfies the second-order ordinary differential equation Q P ′′ + ρP ′ = 0. The most natural choices for the two constants of integration are the saturation point η > 0,and the slope at Q the saturation point, −ζ < 0. With these choices, we have P (Q) = −ζηC η , where 1−ρ −1 z if ρ 6= 1, C(z) = 1−ρ log z if ρ = 1, 8
is the canonical solution with η = ζ = 1. However, the translate the units in which P is measured into the units ζ = 1. This leaves us with 1−ρ η 1 − Qη 1−ρ P (Q) = η(log η − log Q)
only role for the constant ζ is to in which a is measured, so we set
ρ 6= 1,
(16)
ρ = 1.
For ρ < 1, the choke price P (0+ ) = η/(1 − ρ) is finite. For ρ ≥ 1, the choke price is infinite. On (0, η], the pricing curve is convex for ρ > 0, affine for ρ = 0, and concave for ρ < 0. As in the general case, we work with the functions fn (Q) = QP ′ (Q) + nP (Q), but now we have much more precise information about these functions. We define nρ = max(1, ⌊ρ⌋), where ⌊ρ⌋ denotes the largest integer less than or equal to ρ. Lemma 2.8. There is no Nash equilibrium for n < ⌊ρ⌋. For n ≥ nρ , there is a unique solution Q∗n ∈ (0, η) to fn (Q) = An , for all 0 ≤ An < nP (0+ ). Proof. For the specific functional forms arising when ρ is constant, we have fn (Q) = (n + 1 − ρ) P (Q) − η. Hence, when ρ > 1 and n ≤ ρ − 1 < ⌊ρ⌋, then fn < 0 on (0, η), and there is no solution Q∗n to the equation fn (Q) = An for An ≥ 0. When n ≥ nρ , fn is decreasing with fn (η) = −η. If P (0+ ) < ∞, then fn (0+ ) ≥ nP (0+ ) and otherwise fn (0+ ) = ∞. Therefore, there is a unique solution, lying in (0, η), to fn (Q) = An , for 0 ≤ An < nP (0+ ). We can then prove existence by starting from the nρ -player candidate equilibrium and adding players until either (i) no further players wish to enter; or (ii) there are no further players. Of course, we must assume that ρ < N + 1, for otherwise there is no n ≤ N for which an n-player candidate equilibrium exists. The only real obstacle to this program is the possibility that, in an n -player candidate equilibrium, the production level qi of player i is not the global maximum of π(·, Q−i, ai ). Since the case ρ < 2 is already covered by Lemma 2.4, we can restrict attention to the case in which ρ ≥ 2. Moreover, if Q−i were to vanish for some i in an n-player candidate equilibrium that was feasible in all other respects, then that candidate equilibrium would also be a oneplayer candidate equilibrium; and if ρ ≥ 2 then there are no one-player candidate equilibria. We can therefore further restrict attention to the case Q−i > 0. Lemma 2.9. Suppose that 2 ≤ ρ < N + 1, and Q−i > 0. Then g(qi) := π(qi , Q−i , ai ) has a unique global maximum, which is attained in [0, η). Proof. Now g(0) = 0 and g(qi ) < 0 for all qi ∈ [η, ∞), so g has a global maximum attained in [0, η]. From (12), we have qi ′′ g (qi ) = 2 − ρ P ′(Q−i + qi ) Q−i + qi 9
for all qi ∈ [0, ∞). There are then two cases to consider. First, if ρ = 2, then g ′′ < 0 everywhere on [0, so g has a global maximum attained in [0, η). Second, if ρ > 2, h ∞), and 2 Q Q−i 2 Q , ∞ and g is then g ′′ < 0 on 0, ρ−2−i and g ′′ > 0 on ρ−2−i , ∞ . Since g ′′ > 0 on 2ρ−2 h Q−i bounded above, we must have g ′ < 0 on 2ρ−2 , ∞ , and uniqueness follows.
Proposition 2.10. Suppose that ρ < N + 1. Then there is a unique Nash equilibrium given as follows: ¯ ρ Q ∗ qi (a) = max P¯ − ai , 0 , 1 ≤ i ≤ N, η
where
P¯ = min {Pn | nρ ≤ n ≤ N} , and ¯= Q
Pn =
An + η , n+1−ρ
nρ ≤ n ≤ N,
(17)
1 P¯ 1−ρ η 1 − (1 − ρ) if ρ 6= 1, η η exp − P¯ η
if ρ = 1.
The corresponding profits are Gi (a) = qi∗ (a)(P¯ − ai ). In particular, qi∗ and Gi are Lipschitz continuous, and the number of active players in the unique equilibrium is m = min n | nρ ≤ n ≤ N, Pn = P¯ .
Proof. For the constant prudence price curves, the formulas are best expressed in terms of the equilibrium prices. By direct calculation, for each nρ ≤ n ≤ N, the unique solution to fn (Q) = An is given by An + η P (Q∗n ) = =: Pn , (18) n+1−ρ since fn (Q) = (n + 1 − ρ)P (Q) − η. Computing P −1 gives 1 η 1 − (1 − ρ) Pn 1−ρ if ρ 6= 1, η Q∗n = η exp − Pn if ρ = 1. η The candidate n-player equilibria are qi∗
Pn − ai = = −P ′ (Q∗n )
Q∗n η
ρ
(Pn − ai ).
We prove existence and uniqueness as in Proposition 2.6. The only difference is that we now confine attention to nρ ≤ n ≤ N. We start with the nρ -player candidate equilibrium. If nρ = 1, then it is obvious player one will wish to make profit and not leave. If nρ > 1, then 0 < nρ + 1 − ρ ≤ 1 and the candidate price Pnρ is guaranteed to exceed the cost of player nρ : Pnρ =
Anρ + η ≥ Anρ + η > anρ . nρ + 1 − ρ 10
That is, player nρ (and therefore all players i < nρ ) will not want to leave the nρ -player candidate equilibrium. Then Lemma 2.5, Proposition 2.6 and Corollary 2.7 hold as in the general case, except we no longer need the restriction ρ¯ < 2 to guarantee unique global maxima. It remains to characterize the transition point where players stop entering the game. As in Corollary 2.7, this occurs when the equilibrium aggregate production level Q∗n first becomes non-increasing, or we reach the maximum number of players N. This is equivalent to when the candidate prices Pn first become non-decreasing or we reach N, and hence the Nash equilibrium price is given by P¯ . ¯ for some As in the general case, kinks occur in the qi∗ and Gi only when aj = P Q j, that is when player j is exactly indifferent between producing or not. Moreover the qi∗ and Gi are as smooth as P ′ on any region of unit-cost space [0, P (0+))N on which the set {i | qi∗ (a) > 0} of active players is constant. Remark 2.11. We note that the general formula for Pn (obtained without setting ζ = 1) is (An +ζ η)/(n+1−ρ), which resolves the seemingly inconsistent dimensions in the numerator of (17). Remark 2.12. There is an even more explicit characterization of m. Consider the function h given by the formula h(n) = η + An−1 − (n − ρ) an . The requirement that a1 < P (0+ ) ensures that h(1) = η − (1 − ρ) a1 > 0. Moreover h(n + 1) − h(n) = (ρ − n) (an+1 − an ). Hence h(n + 1) − h(n) ≥ 0 if and only if n ≤ ρ. Then m is the largest n ∈ {1, 2, ..., N} such that h(n) > 0. In the numerical solutions in Section 5.3, we shall use the linear price function corresponding to ρ = 0, that is, P (Q) = η − Q. In this case, the market price is An + η ¯ P = min |1≤n≤N ; (19) n+1 the quantity produced by player i is qi∗ (a) = max P¯ − ai , 0 for all 1 ≤ i ≤ N; and the 2 profit of player i is Gi (a) = qi∗ (a)(P¯ − ai ) = max P¯ − ai , 0 . Alternatively, in terms of the number of active players m = min{1 ≤ n ≤ N | Pn = P¯ }, we have ! m X 1 aj , Gi (a) = (qi∗ (a))2 , qi∗ (a) = η − mai + m+1 j=1,j6=i for 1 ≤ i ≤ m, and qi∗ = Gi = 0 for i > m. Finally, we mention that ordering of the players by costs is, of course, not crucial to defining the functions qi∗ and Gi in this section. Given a general costs vector, the constructions above are simply modified to first temporarily relabel the firms according to their costs, compute the Nash equilibrium as above, and then return the equilibrium quantities and profits in the original labelling order.
11
3
Differential Game & Exhaustibility
We now introduce the dynamic Cournot game under exhaustibility constraints. Each player i has reserves of a traditional and cheap-to-produce resource (for example oil, by extraction), denoted by xi (t) at time t ≥ 0. We take for simplicity the cost of production from this source to be zero, but reserves are finite (exhaustible). There is also an alternative source that is inexhaustible, but expensive-to-produce (solar power in the energy example), with constant unit cost of production c ∈ [0, P (0+ )). Player i chooses a dynamic production rate q¯i that is a Markov strategy:1 q¯i = q¯i (x(t)), where x(t) = (x1 (t), · · · , xN (t)). As long as xi > 0, player i has the choice between producing from the cheap or expensive sources. After xi hits zero, he can only produce from the costlier alternative, which never runs out. We shall suppose, at first, that no player produces from the more expensive source as long as the cheaper one is available,2 and we will discuss how this could be validated a posteriori. Therefore, reserves of his traditional resource deplete according to dxi = −¯ qi (x(t)), xi > 0. dt (To lighten the notation, we do not denote the dependence of x on the q¯i .) The market price is governed by a Cournot competition with the price function P as before, satisfying Assumption 2.2. We assume that, given a cost vector a satisfying (3), there is a unique Nash equilibrium q ∗ (a) of the static Cournot game. Some general conditions for this were given in Proposition 2.6, and, for a specific family of price functions, in Proposition 2.10.
3.1
Dynamic Cournot Competition
Given initial reserves xi (0) ≥ 0, player i wants to maximize his discounted lifetime profit Z ∞ ¯ −i (x(t)), c1I{x (t)=0} dt, e−rt π q¯i (x(t)), Q i 0
where 1I denotes the indicator P function, r > 0 is a discount rate, the profit function π was ¯ defined in (1), and Q−i = j6=i q¯j . Note that the cost of production rises from zero to c when reserves xi run out, as denoted in the third argument of π in the integral. ∗ We look for a Markov Perfect Nash equilibrium q¯ ∗ (x(t)) = (¯ q1∗ (x(t)), · · · , q¯N (x(t))) such ∗ that, for each player i, and each initial state x(0), q¯i is the best response whenPall the other ¯∗ = players play their equilibrium strategies. Therefore, with the notation Q ¯j∗ , −i j6=i q Z
∞
−rt
e
0
π
¯ ∗−i (x(t)), c1I{x (t)=0} q¯i∗ (x(t)), Q i
dt ≥
Z
0
∞
¯ ∗−i (x(t)), c1I{x (t)=0} dt, e−rt π q¯i (x(t)), Q i
N for any Markov strategy q¯i of player i, and for all x(0) ∈ RN + = [0, ∞) . The requirement that the equilibrium strategies are independent of the initial resource levels x(0) is equivalent, 1
Markov strategies are also sometimes called feedback or closed-loop strategies. According to the lead editorial in The Times of London on 13 July, 2009: “No sane energy company would, while fossil fuels are still plentiful, voluntarily opt for a more expensive, less reliable energy source.” 2
12
in our setting, to the requirement that the equilibrium is perfect, sometimes called subgame perfect. This excludes equilibria with so-called “incredible threats” whereby players may make extreme, but unrealistic, threats of increased production if another player deviates from a certain path. We refer to [14] and the textbooks [9, 10] for further discussion and references on this issue. We give an informal motivation for the dynamic programming PDEs we shall use to construct Nash equilibria for these problems. First, consider any continuous Markov perfect strategy {¯ qj (x(t)) | t ≥ 0, 1 ≤ j ≤ N}, and the profits starting at time s ≥ 0: Z ∞ q¯ ¯ −i (x(t)), c1I{x (t)=0} dt, 1 ≤ i ≤ N. vi (x(s)) = e−r(t−s) π q¯i (x(t)), Q (20) i s
Let x = x(0) denote any interior point (xj > 0, for all 1 ≤ j ≤ n), so all players start with some intitial reserves.3 Then, differentiating (20) with respect to s and setting s = 0 gives the partial differential equation ¯ −i (x), 0) − π(¯ qi (x), Q
N X
∂viq¯ q¯j (x) − rviq¯ = 0, ∂xj j=1
which can be re-written as X ∂viq¯ ∂viq¯ ¯ q¯j (x) − − rviq¯ = 0. π q¯i (x), Q−i (x), ∂xj ∂xj j6=i
(21)
(22)
Then Bellman-principle arguments4 reduce each player’s optimization problem to a local optimization, and the search for a Markov perfect equilibrium to a search for a local static Nash equilibrium, which, in this case, amounts to optimizing π in (22) with respect to its first argument, ∗with the second argument fixed at the other players’ equilibrium strategies. Writing vi = viq¯ for the value functions using the equilibrium policies: Z ∞ ¯ ∗ (x(t)), c1I{x (t)=0} dt, vi (x) = e−rt π q¯i∗ (x(t)), Q −i i 0
we have
X ∂vi ∂v i ∗ ¯ (x), − q¯j∗ (x) max π qi , Q − rvi = 0, −i qi ≥0 ∂xi ∂xj j6=i
i = 1, · · · , N.
(23)
where x ∈ RN + with xj > 0 for 1 ≤ j ≤ N, that is when all players have some reserve of the traditional resource. We assume throughout that each vi is continuously differentiable up to the axes: vi ∈ ∂vi C 1 (RN ¯i∗ (x) is continuous at all x ∈ RN + ); and that q + . We observe from (23) that ∂xi enters as a “shadow cost” for player i at the differential level. The interpretation as a cost is ∂vi legitimate as we naturally expect ∂x ≥ 0: adding more reserves increases the value function. i 3 4
The boundary cases when some xj = 0 are dealt with in Section 3.2. See, for instance, [15], [8, Section 8.2], [1, Section 6.5.2], or [6, Section 4.2].
13
Comparison with (2) reveals the differential Nash equilibrium problem in the PDEs is just the one-period game with the role of the costs ai played by the partial derivatives ∂vi /∂xi . ∗ For a fixed x ∈ RN + , if there is a unique Nash equilibrium q (a) for the static game with ai =
∂vi (x), ∂xi
i = 1, · · · , N,
then we re-write equations (23) as X ∂vi − rvi = 0, Gi (Dv) − qj∗ (Dv) ∂xj j6=i
where we define
i = 1, · · · , N,
(24)
∂v1 ∂vN Dv = diag(∇v) = ,··· , , ∂x1 ∂xN and recall that Gi (a) = qi∗ (a)(P (Q∗ ) − ai ) is equilibrium profit function of the static game. The equilibrium production rates of the exhaustible resource at time t are given by q¯i∗ (x(t)) = qi∗ (Dv(x(t))). Note that the definition of Nash equilibrium and the constructions of qi∗ (a) and Gi (a) in Propositions 2.6 and 2.10 take care of the fact that not all players may participate at all resource levels x, depending on the vector of shadow costs a = Dv(x) at that point. However they encompass the fact that there are always potentially N active players. For the majority of the paper, we shall treat the cases where all players participate, but we shall discuss the situation where one player may be blockaded in the two-player dynamic game in Section 5. At this level of generality, we are not able to provide reasonable conditions for existence and uniqueness of a solution to the system (24), equipped with appropriate boundary conditions discussed in the following section, let alone solutions with sufficient regularity to generate a unique Nash equilibrium with well-behaved strategies. We will proceed by staying relatively close to a case which is well-understood. In the next section, we address the issue of exhaustibility and boundary conditions.
3.2
Exhaustibility
When a player has exhausted his reserves of the cheap resource, he can turn to the alternative means of production which, while more costly than the original one, allows the exhausted player to remain in the game, but in a disadvantaged position. In the energy example, there are alternative “backstop” technologies, such as solar power or steam-extracted oil shales, that an energy supplier may resort to when his reserves of oil run out, both of which are more expensive than delivering energy by extracting oil. We consider the case xi = 0, when player i has exhausted his supply. Then we have dxi = 0. dt The HJ equation for vi (x1 , · · · , xi−1 , 0, xi+1 , · · · , xN ) becomes X ∗ ∂vi ¯ ∗ (x), c − qj (Dv) − rvi = 0, max π qi , Q −i qi ≥0 ∂xj j6=i 14
or Gi (D−i v) − where we define D−i v =
X
qj∗ (D−i v)
j6=i
∂vi − rvi = 0, ∂xj
∂vi−1 ∂vi+1 ∂vN ∂v1 ,··· , , c, ,··· , ∂x1 ∂xi−1 ∂xi+1 ∂xN
(25)
.
Similarly, the HJ equation for vj (x1 , · · · , xi−1 , 0, xi+1 , · · · , xN ), j 6= i becomes Gj (D−i v) −
X
qk∗ (D−i v)
k ∈{i,j} /
∂vj − rvj = 0. ∂xk
(26)
These are then a system of N equations on the hyperplane xi = 0 involving only partial derivatives along the plane. We can proceed similarly to the cases in which more and more players have exhausted their reserves until we reach the fully exhausted case. Here, all the players are using the inexhaustible alternative resource, and so produce at the constant rate qi∗ (c1), where 1 denotes the N-vector of ones. It follows that the value functions are just the constant Gi (c1) . (27) vi (0) = r This serves as the initial condition for the one-player ODEs on the lines. Once solved, these are axis Dirichlet boundary conditions for the two-player PDEs on the planes, and so on. Now if the value functions are found to satisfy ∂vi < c in {xi > 0}, ∂xi then our initial hypothesis, that no producer will use the alternative technology while the cheaper one is available, will be validated. This is clear because if we introduced additional control variables in the interior so that each player could choose both a quantity from reserves and a quantity from the alternative source, then in each game at the differential level, none ∂vi would be produced from the alternative as long as the shadow cost ∂x is smaller than c. This i is indeed the case for the approximate solutions in Section 4.2, and the numerical solutions of Section 5.3, and we shall assume it to be the case for the remainder of the paper.
3.3
Neumann Boundary Conditions
When all players participate at all resource levels, it is possible to replace the Dirichlet boundary conditions coming from (26) by simpler Neumann boundary conditions. We reiterate that the Dirichlet conditions for the value functions are found by solving the chain of games starting with the case of all players exhausted, then the game where one has reserves and N − 1 players are using the alternative technology, up to N − 1 with reserves and one using the alternative. Recall that we have assumed that we have continuity of the first derivatives of the value functions up to the boundaries and continuity of the equilibrium policies q¯i∗ (x) at all x ∈ RN +.
15
ˆ denote the projection of x Fix 1 ≤ i ≤ N and a point x ∈ RN + with all xj > 0, and let x onto {xi = 0}. For player i, his equilibrium production on xi = 0 is given by ∂vi−1 ∂vi+1 ∂vN ∂v1 ∗ ∗ ∗ (ˆ x), · · · , (ˆ x), c, (ˆ x), · · · , (ˆ x) , q¯i (ˆ x) = qi (D−i v(ˆ x)) = qi ∂x1 ∂xi−1 ∂xi+1 ∂xN whereas in the interior it is q¯i∗ (x)
=
qi∗ (Dv(x))
=
qi∗
∂v1 ∂vN (x), · · · , (x) . ∂x1 ∂xN
ˆ . In general, we expect that From continuity of q¯i∗ (x), qi∗ (Dv(x)) → qi∗ (D−i v(ˆ x)) as x → x ∗ qi (a) is a continuous function, which was true in the cases of Corollary 2.7 and Proposition 2.10. Further, we expect the static equilibrium quantity qi∗ (a) to be decreasing in its i-th argument, as long as qi∗ > 0. Specifically, if it is the case that ∂qi∗ (D−i v(ˆ x)) < 0 and qi∗ (D−i v(ˆ x)) > 0, ∂ai then we can conclude that
∂vi = c on xi = 0. (28) ∂xi The interpretation of this expression is that, on hitting the boundary xi = 0, the shadow cost of player i turns into the real cost c. As long as player i participates in the game on xi = 0, then we have the Neumann boundary condition (28). Comparison of (26) with (24), which we re-write as Gj (Dv) −
X
qk∗ (Dv)
k6=j
∂vj − rvj = 0, ∂xk
yields
∂vj = 0 on xi = 0, j 6= i (29) ∂xi provided qi∗ (D−iv) 6= 0. Therefore, as long as player i still participates in the game on xi = 0, when he is forced to use the alternative technology, the shadow cost of the other players is zero. Indeed, when the cost c is small enough, all players will participate at all resource levels. For example, in the case of the constant prudence price curves of Section 2.2, we need only consider the extreme case where there are N − 1 producers with the minimum unit cost of zero, and one with the maximum possible unit cost c. If ρ ∈ [N, N + 1), then only a full N-player Nash equilibrium is possible. As the candidate market price from (17) is PN =
η+c > c, N +1−ρ
since 0 < N + 1 − ρ ≤ 1, all players participate in this case for any c < ∞. In the case ρ < N, the candidate market price PN −1 = η/(N − ρ) exceeds c if c
0. We remark that the inexhaustible limit (31) is also the behaviour for large discounting rate r or large resources. Indeed if we write the value functions as vi (x; r) to stress the discount rate, it is easy to check from the PDEs (24) and boundary conditions (28) and (29), that 1 vi (x; r) = vi (rx; 1), r so formal asymptotics in the limits of large (or small) r or ||x|| are analogous calculations.
4
Small Exhaustibility Approximation
When c is small, we are close to the inexhaustible game played repeatedly, and we may expect that all players participate at all resource levels. In preparation for an approximation in this case for the dynamic game, we first analyze the effect of small-costs on the static game.
4.1
Static Game Small Cost Perturbation
We return to the static game, whose Nash optimal strategies are given by qi∗ (a), and equilibrium profits are Gi (a) = qi∗ (a)(P (Q∗ ) − ai ), (32) where Q∗ satisfies (7). We assume costs a are such that all players participate in the equilibrium and the qi∗ (a) (and hence the Gi (a)) are differentiable. A small costs Taylor expansion gives X Gi (a) ≈ Gi (0) + Aai + B aj , j6=i
where we define the constants
A=
∂Gi (0), ∂ai
B=
17
∂Gi (0), ∂aj
j 6= i,
(33)
which are independent of i and j. Similarly, the strategies are given approximately by X aj , (34) qi∗ (a) ≈ γ + λai + µ j6=i
where we define γ=
∂qi∗ λ= (0), ∂ai
qi∗ (0),
∂qi∗ µ= (0), ∂aj
j 6= i,
(35)
which are again independent of i and j. Further, we define the constant of relative prudence at the zero cost equilibrium solution by ρ0 = −Nγ
P ′′ (Nγ) , P ′(Nγ)
(36)
which is just ρ(Nγ), where ρ(q) was defined in (4). Then we have the following expressions for (A, B, λ, µ) in terms of (γ, ρ0 , P ′(Nγ)). Proposition 4.1. We assume ρ0 < (N + 1). The perturbation coefficients (A, B, λ, µ) can be expressed as 2N − (2 − N −1 )ρ0 A = −γ , (37) (N + 1) − ρ0 2 − N −1 ρ0 B = γ , (38) (N + 1) − ρ0 N − (1 − N −1 )ρ0 1 , (39) λ = P ′ (Nγ) (N + 1) − ρ0 1 − N −1 ρ0 1 . (40) µ = − ′ P (Nγ) (N + 1) − ρ0 Proof. Let θ = λ + (N − 1)µ. Differentiating the summed first-order conditions (7) with respect to ai and setting a = 0 gives (N + 1)θP ′ (Nγ) − 1 + NγθP ′′ (Nγ) = 0, which gives θ=
1 1 = . (N + 1)P ′ (Nγ) + NγP ′′ (Nγ) P ′ (Nγ)((N + 1) − ρ0 )
Differentiating (6) with respect to ai and setting a = 0 gives θP ′ (Nγ) − 1 + λP ′ (Nγ) + γθP ′′ (Nγ) = 0, which yields λ=
1 P ′ (Nγ)
(1 − θ(P ′ (Nγ) + γP ′′ (Nγ))).
Re-arranging leads to (39) and (40). 18
Next, differentiating (32) with respect to ai and setting a = 0 gives
But, from (7) with AN =
P
A = λP (Nγ) + γ(θP ′(Nγ) − 1).
j
aj = 0, we have P (Nγ) = −γP ′ (Nγ). Therefore,
A = γ(θ − λ)P ′ (Nγ) − γ = γ(N − 1)µP ′ (Nγ) − γ, and (37) follows after substituting from (40). Similarly, differentiating (32) with respect to aj (j 6= i) and setting a = 0 gives B = γ(θ − µ)P ′(Nγ), and (38) follows after substitution for θ and µ. We comment that since ρ0 < (N + 1) (and N ≥ 2), the formulas (38-40) imply that B > 0,
and λ < 0,
(41)
while (37) implies that A ≤ 0 for ρ0 ≤ and A > 0 otherwise. Formula (40) yields
2N 2 , 2N − 1
(42)
µ ≥ 0 for ρ0 ≤ N, and µ < 0 otherwise. In other words, player i’s profits increase when any other player’s cost aj is increased from zero, and he also increases his production for ρ0 ≤ N. When his own cost ai is increased from zero, he decreases his production, but his profit may increase or decrease depending on ρ0 . Finally, if costs for all the players are increased by the same amount, ai = c, ∀i, then Gi (c1) ≈ Gi (0) + (A + (N − 1)B)c, so each player’s equilibrium profit increases with cost according to the sign of A + (N − 1)B. From (37) and (38), we have A + (N − 1)B = γ
(ρ0 − 2) , (N + 1) − ρ0
so profits actually increase with costs for ρ0 > 2, suggesting that sufficient curvature induces a degree of mutually beneficial production. Similarly, the optimal production quantities under a symmetric increase in costs are approximated as qi∗ (c1) ≈ qi∗ (0) + (λ + (N − 1)µ)c. From (39) and (40), λ + (N − 1)µ =
1 P ′ (Nγ)((N
+ 1) − ρ0 )
< 0,
(43)
so each player’s equilibrium production decreases with an across-the-board cost increase. 19
4.2
Differential Game Small Cost Perturbation
We look for an approximate solution of (24), with boundary conditions (28)-(29), of the form vi (x) =
Gi (0) (1) + cvi (x) + o(c), r
(1)
for some functions vi to be found. Inserting the expansion into the PDEs and boundary conditions leads to the linearized system (1)
(1)
A
(1)
X ∂vj X ∂v ∂vi (1) i +B −γ − rvi = 0, ∂xi ∂xj ∂xj j6=i
(44)
j6=i
(1)
∂vi |x =0 = 1, ∂xi i (1) ∂vi |x =0 = 0 (j 6= i), ∂xj i
(45) (46)
where (A, B, γ) were defined in (33) and (35). We have the following explicit solution. Proposition 4.2. Assume that ρ0
